(2015) Chemical Kinetics by M Lim 1

20.9 The diffusion equation

( ) ( )

( ) ( )

2

2

1

J x Adt J x l Adtct Al dt

J x J x l Jl x

c cD Dx x x

− +∂=

∂− + ∂

= = −∂

∂ ∂ ∂ = − − = ∂ ∂ ∂

2

2 (Fick's 2nd law)c cDt x

∂ ∂=

∂ ∂

Time-dependent diffusion process:

can be used to predict c(x,t)

(2015) Chemical Kinetics by M Lim 2

20.9 (b) solutions of the diffusion equation2

2

c cDt x

∂ ∂=

∂ ∂2

0 41Solution for 1-d, ( , )xDtnc x t e

A Dtπ−

=

n0 x

Dt=

2

22

2

1Gaussian, ( )2

2 4 2

x

G x e

Dt Dt

σ

σ

πσ

σ

−=

→ ==

( )

2

2

40 3 2

for 3-d

1If spherically symmetric, ( , )8

rDt

c D ct

c r t n eDtπ

−

∂= ∇

∂

=

A

(2015) Chemical Kinetics by M Lim 3

Review 20-5

[ ],

1 (force arising from gradient)

(Stokes-Einstein relation)

, (Einstein relation)

p T

cRTc x

kTDf

zFD uRTu DRT zF

∂ = − ∂

=

= =

F

2

2

c cDt x

∂ ∂=

∂ ∂

zuFλ =

2

0 41Solution for 1-d, ( , )xDtnc x t e

A Dtπ−

=

2 for 3-dc D ct

∂= ∇

∂

( )

2

40 3 2

1If spherically symmetric, ( , )8

rDtc r t n e

Dtπ

− =

(2015) Chemical Kinetics by M Lim 4

20.9 (a) Diffusion with convection

Convective flux, ( : flowing velocity)

cAv tJ cv vA t

c J cvt x x

∆= =

∆∂ ∂ ∂

= − = −∂ ∂ ∂

Transport by the motion of a streaming fluid

2

2 (contribution from reaction)c c cD vt x x

∂ ∂ ∂= − +

∂ ∂ ∂Used in reactor design,

the utilization of resources in living cells

• Measurement of D: capillary techniquediaphragm technique

Generalized diffusion eqn (diffusion with convection)

(2015) Chemical Kinetics by M Lim 5

20.10 Diffusion probabilitiesThe solution of the diffusion equation can be used to predict c(x,t), and to calculate the net distance through which the particles diffuse in a given time

The average distance travelled by a diffus

2

ing par

(see ne

ticl

xt s

e

lide)Dtxπ

=

When particles diffuse in both directions,<x>=0 at all times

10 2 1

2

10 10 2

2 5 1

7 5

6 3

4 1

3 3

When 5 10

For 0.1 ,

2 5 10 0.1 10

10 10

10 10 10 10 10 10 10 10

D m s

s x

m

x m for s

m for sm for sm for sm for s

− −

− −

− −

− −

− −

−

−

= ×

= ⋅ × ⋅ =

=

( )2

0

2

2 2

2x

x x

x D

P x Dt

tσ

∞

=

=

=

=∫

( )2

0

2

2 24 6

6 r

r r P r r

r

r

D

d Dt

t

π

σ

∞=

= =

=∫For 3D

(2015) Chemical Kinetics by M Lim 6

20.10 Diffusion probabilities (Ex 20.5)

x0

2

2

2

0

0

2 3 2

0

12

12

41

4

ax

ax

ax

e dxa

xe dxa

x e dx a

aDt

π

π

∞ −

∞ −

∞ − −

=

=

=

=

∫

∫

∫( )

( )

2

2

0 0

0

0 00

4

0 40

0

The number of particles in a slab: Probability that any of the particles in the slab:

=

1

1

A

A

A

A

xA Dt

xDt

cN AdxN n N

cN AdxP x dx dxN

cN Ax xP x x dxN

N A nx e dxN A Dt

eDt

π

π∞ ∞

∞ −

−

=

=

= =

=

=

∫ ∫

∫

( )

2

2

40

2 2 2 40 0

1

2

2

xDt

xDt

xe dxDt

x x P x

Dt

x Dte dxDt

π π

π

∞ −

∞ ∞ −

=

= = =

∫

∫ ∫

2

0 4

Solution for 1-d,

1( , )xDtnc x t e

A Dtπ−

=

(2015) Chemical Kinetics by M Lim 7

20.10 Diffusion probabilities for 3D2

2

2

2

2

0

0

2 3 2

0

320

4 5 2

0

12

12

41

23

4 21

4

ax

ax

ax

ax

ax

e dxa

xe dxa

x e dx a

x e dxa

x e dx a

aDt

π

π

π

∞ −

∞ −

∞ − −

∞ −

∞ − −

=

=

=

=

=

=

∫

∫

∫

∫

∫( )

( )

( )( )

( )( )

2

2

2

43 2

2 243 20 0

2 2 2 2 243 20 0

,

18

14 4 48

14 48

6

π

π πππ

π ππ

−

∞ ∞ −

∞ ∞ −

=

= = =

= = =

∫ ∫

∫ ∫

rDt

rDt

rDt

For spherically symmetric case

P r eDt

Dtr rP r r dr re r drDt

r r P r r dr r e r drDt

Dt

(2015) Chemical Kinetics by M Lim 8

20.10 Diffusion probabilities for 2D2

2

2

2

2

0

0

2 3 2

0

320

4 5 2

0

12

12

41

23

4 21

4

ax

ax

ax

ax

ax

e dxa

xe dxa

x e dx a

x e dxa

x e dx a

aDt

π

π

π

∞ −

∞ −

∞ − −

∞ −

∞ − −

=

=

=

=

=

=

∫

∫

∫

∫

∫( )

( )

( )

2

2

2

4

40 0

2 2 2 40 0

,

14

12 24

4412 2

π

π π ππ

π ππ

−

∞ ∞ −

∞ ∞ −

=

= = =

= = =

∫ ∫

∫ ∫

rDt

rDt

rDt

For radially symmetric case

P r eDt

r rP r rdr re rdr DtDt

r r P r rdr r e rdrDt

Dt

(2015) Chemical Kinetics by M Lim 9

20.11 The statistical view

( )2

222 xtP x e

t

τλτ

π−

=

2 2

For a perfect gas, ,

and : mean free path 1 1

2 2 2 3

c

cD c c

λτ

λ

λ λ λ λτ λ

=

= = = →

Random walk: particles jump through a distance λ (step length) in a time τ (duration)

The probability of a particle being at a distance x from the origin after a time tin 1-d random walk.

( )2 2

22 4

2

2

12 4

x xt DtP x e e

t

t Dt

τλτ

πτλ

− −= ∝

=

Einstein-Smoluchowski equationLink between the microscopic details of particle motion and the macroscopic parameters relating to diffusion.

Illus. For SO42-, D=1.1×10-9 m2 s-1, a=210 pm

Suppose it jumps through its own diameter, λ=2a, then τ=80 ps,meaning that 1×1010 jumps per second.

2

2D λ

τ=

(2015) Chemical Kinetics by M Lim 10

Review 20-6[ ]

,

1 (force arising from gradient)

(Stokes-Einstein relation)

, (Einstein relation)

p T

cRTc x

kTDf

zFD uRTu DRT zF

∂ = − ∂

=

= =

F

2

2 (contribution from reaction)c c cD vt x x

∂ ∂ ∂= − +

∂ ∂ ∂2 2 for 1D

4 for 2D

6 for 3D

x Dt

Dt

Dt

=

=

=( )

2

22

22 2

xtP x e D

t

τλτ λ

π τ−

= =

(2015) Chemical Kinetics by M Lim 11

20.11 The statistical view-1

( ) ( )

( ) ( )

(1-d) and

The net distance after steps: ( ) ,, net distance of travel:

1 1Since , and 2 2

! !The number of way to arrive at : ( )1 1! ! ! !2 2

R L

R L

R L R L

R L

tN

N N Nn N N x n

N N N N N n N N n

N Nx W xN N N n N n

τλλ

=

−≡ − =

− = = + = −

= = + −

( ) ( )

Toral number of step: 2! !The probability of the net distance walked being : ( )

1 1! ! 2 ! !2 2

1For large , ln ! ln - ln 2 (Stirling's approximation)2

ln ( )

N

NR L

N Nx P xN N N n N n

N x x x x

P x

π

= = + −

≈ + +

( ) ( )1 1ln ! ln ! ln ! ln 22 2

N N n N n N = − + − − −

(2015) Chemical Kinetics by M Lim 12

20.11 The statistical view-2( ) ( ) ( )

( ) ( ) ( )

1 1 1 1 1ln ( ) ln ln 2 ln ln 22 2 2 2 2

1 1 1 1 ln ln 2 ln 22 2 2 2

1 2

P x N N N N n N n N n

N n N n N n N

N

π π

π

= + − + − + + + − + +

− − + − − − + −

= +

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( )

1ln 1 ln 1 ln 221 1 ln 1 ln 2 ln 2 ln 22

1 1 1 ln 1 ln 1 1 ln2 2 2

1 1 ln 12

N N n N n N n

N n N n N n N

nN N N n N n NNnN nN

π

− + + + + + +

− − + − + − + − −

= + − + + + − + +

− − + − ( )

( ) ( )

( ) ( )

( ) ( )2

2

1 1 ln ln 2 ln 22

1 1 1 ln 2 ln ln 2 1 ln 1 1 ln 12 2 2

2 1 1 ln 1 ln 1 1 ln 12 2

2 1 1 1 1 ln 1 12 2 2 2

N n N

n nN N n N nN N

n nN n N nN N N

n n n nN n N nN N N N

π

π

π

π

− − + − −

= − − − + + + − − + −

= − + + + − − + −

− + + − − + − − −

2

2N

(2015) Chemical Kinetics by M Lim 13

20.11 The statistical view-3( )( ) ( )( )

( ) ( )

( ) ( )

( )

2 2

2 2

2 2 3

2 2

2 2 3

2 2

2 2

2

2 1 1 1 1ln ( ) ln 1 12 2 2 2

1 11 12 22 1 ln

2 1 11 12 2

2 1 1 ln 12 2

n n n nP x N n N nN N N N N

n n n nN NN N N N

N n n n nN NN N N N

n nNN N N

π

π

π

= − + + − − + − − −

+ − + + − = −

+ − + − + + +

= − − + + +

( )

( )

2

2

2

2 2

2

2 2 2 2 2

2 2

2

2

2

1 12

2 1 2 2 1 2 ln 1 ln2 2

2ln ( ) ln2

2( ) and

2( )

nN

xt

n nNN N

n n n n nNN N N N N N N

nP xN N

x tP x e n NN

P x et

τλ

π π

π

π λ τ

τπ

−

−

− + +

= − − + + = − − − +

≈ −

= = =

∴ =